Joceline C Lega

Interests

Teaching

Calculus, Linear Algebra, Differential Equations, Dynamical Systems, Real Analysis, Partial Differential Equations, Mathematical Modeling, Methods of Applied Mathematics, Integrated Science.

Research

Modeling of nonlinear phenomena, with applications to physics and biology. Pattern formation and instabilities.Dynamics and stability of coherent structures.

Courses

Scholarly Contributions

Journals/Publications

• Ercolani, N., Lega, J., & Tippings, B. (2023). Multiple Scale Asymptotics of Map Enumeration. Nonlinearity, 36, 1663-1698. doi:https://doi.org/10.1088/1361-6544/acb47d
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  We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann-Hilbert analysis of orthogonal polynomials and the theory of discrete dynamical systems. By equating the coefficients of these expansions in a common region of uniform validity in their parameters, we recover known results and provide new expressions for generating functions associated with graphical enumeration on surfaces of genera 0 through 7. Although the body of the article focuses on 4-valent maps, the methodology presented here extends to regular maps of arbitrary even valence and to some cases of odd valence, as detailed in the appendices. [Journal_ref: Nonlinearity 36, 1663-1698 (2023)]
• Ercolani, N., Lega, J., & Tippings, B. (2023). Non-recursive Counts of Graphs on Surfaces. Enumerative Combinatorics and Applications, 3(3), 1-24. doi:https://doi.org/10.54550/ECA2023V3S3R20
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  The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of the surface. As such this problem lies at the cross-roads of combinatorial studies in low dimensional topology and graph theory. The determination of explicit formulae for map counts, in terms of closed classical combinatorial functions of $g$ and $j$ as opposed to a recursive prescription, has been a long-standing problem with explicit results known only for very low values of $g$. In this paper we derive closed-form expressions for counts of maps with an arbitrary number of even-valent vertices, embedded in surfaces of arbitrary genus. In particular, we exhibit a number of higher genus examples for 4-valent maps that have not appeared prior in the literature.
• Cramer, E. Y., Huang, Y., Wang, Y., Ray, E. L., Cornell, M., Bracher, J., Brennen, A., Rivadeneira, A. J., Gerding, A., House, K., Jayawardena, D., Kanji, A. H., Khandelwal, A., Le, K., Mody, V., Mody, V., Niemi, J., Stark, A., Shah, A., , Wattanchit, N., et al. (2022). The United States COVID-19 Forecast Hub dataset. Scientific Data, 9, 462. doi:https://doi.org/10.1038/s41597-022-01517-w
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  Academic researchers, government agencies, industry groups, and individuals have produced forecasts at an unprecedented scale during the COVID-19 pandemic. To leverage these forecasts, the United States Centers for Disease Control and Prevention (CDC) partnered with an academic research lab at the University of Massachusetts Amherst to create the US COVID-19 Forecast Hub. Launched in April 2020, the Forecast Hub is a dataset with point and probabilistic forecasts of incident cases, incident hospitalizations, incident deaths, and cumulative deaths due to COVID-19 at county, state, and national, levels in the United States. Included forecasts represent a variety of modeling approaches, data sources, and assumptions regarding the spread of COVID-19. The goal of this dataset is to establish a standardized and comparable set of short-term forecasts from modeling teams. These data can be used to develop ensemble models, communicate forecasts to the public, create visualizations, compare models, and inform policies regarding COVID-19 mitigation. These open-source data are available via download from GitHub, through an online API, and through R packages.
• Cramer, E. Y., Ray, E. L., Lopez, V. K., Bracher, J., Brennen, A., Castro Rivadeneira, A. J., Gerding, A., Gneiting, T., House, K. H., Huang, Y., Jayawardena, D., Kanji, A. H., Khandelwal, A., Le, K., Mühlemann, A., Niemi, J., Shah, A., Stark, A., Wang, Y., , Wattanachit, N., et al. (2022). Evaluation of individual and ensemble probabilistic forecasts of COVID-19 mortality in the United States. Proceedings of the National Academy of Sciences of the United States of America, 119(15), e2113561119. doi:https://doi.org/10.1073/pnas.2113561119
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  Short-term probabilistic forecasts of the trajectory of the COVID-19 pandemic in the United States have served as a visible and important communication channel between the scientific modeling community and both the general public and decision-makers. Forecasting models provide specific, quantitative, and evaluable predictions that inform short-term decisions such as healthcare staffing needs, school closures, and allocation of medical supplies. Starting in April 2020, the US COVID-19 Forecast Hub (https://covid19forecasthub.org/) collected, disseminated, and synthesized tens of millions of specific predictions from more than 90 different academic, industry, and independent research groups. A multimodel ensemble forecast that combined predictions from dozens of groups every week provided the most consistently accurate probabilistic forecasts of incident deaths due to COVID-19 at the state and national level from April 2020 through October 2021. The performance of 27 individual models that submitted complete forecasts of COVID-19 deaths consistently throughout this year showed high variability in forecast skill across time, geospatial units, and forecast horizons. Two-thirds of the models evaluated showed better accuracy than a naïve baseline model. Forecast accuracy degraded as models made predictions further into the future, with probabilistic error at a 20-wk horizon three to five times larger than when predicting at a 1-wk horizon. This project underscores the role that collaboration and active coordination between governmental public-health agencies, academic modeling teams, and industry partners can play in developing modern modeling capabilities to support local, state, and federal response to outbreaks.
• Ercolani, N., Lega, J., & Tippings, B. (2022). Dynamics of Non-polar Solutions to the Discrete Painlevé I Equation. SIAM Journal on Applied Dynamical Systems, 21, 1322-1351. doi:https://doi.org/10.1137/21M1445156
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  This manuscript develops a novel understanding of non-polar solutions of the discrete Painleve I equation (dP1). As the non-autonomous counterpart of an analytically completely integrable difference equation, this system is endowed with a rich dynamical structure. In addition, its non-polar solutions, which grow without bounds as the iteration index $n$ increases, are of particular relevance to other areas of mathematics. We combine theory and asymptotics with high-precision numerical simulations to arrive at the following picture: when extended to include backward iterates, known non-polar solutions of dP1 form a family of heteroclinic connections between two fixed points at infinity. One of these solutions, the Freud orbit of orthogonal polynomial theory, is a singular limit of the other solutions in the family. Near their asymptotic limits, all solutions converge to the Freud orbit, which follows invariant curves of dP1, when written as a 3-D autonomous system, and reaches the point at positive infinity along a center manifold. This description leads to two important results. First, the Freud orbit tracks sequences of period-1 and 2 points of the autonomous counterpart of dP1 for large positive and negative values of $n$, respectively. Second, we identify an elegant method to obtain an asymptotic expansion of the iterates on the Freud orbit for large positive values of $n$. The structure of invariant manifolds emerging from this picture contributes to a deeper understanding of the global analysis of an interesting class of discrete dynamical systems.
• Sahneh, F. D., Fries, W., Watkins, J. C., & Lega, J. (2022). Epidemics from the Eye of the Pathogen. SIAM J. Appl. Math., 82, 2036-2056. doi:https://doi.org/10.1137/21M1450719
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  While a common trend in disease modeling is to develop models of increasing complexity, it was recently pointed out that outbreaks appear remarkably simple when viewed in the incidence vs. cumulative cases (ICC) plane. This article details the theory behind this phenomenon by analyzing the stochastic SIR (Susceptible, Infected, Recovered) model in the cumulative cases domain. We prove that the Markov chain associated with this model reduces, in the ICC plane, to a pure birth chain for the cumulative number of cases, whose limit leads to an independent increments Gaussian process that fluctuates about a deterministic ICC curve. We calculate the associated variance and quantify the additional variability due to estimating incidence over a finite period of time. We also illustrate the universality brought forth by the ICC concept on real-world data for Influenza A and for the COVID-19 outbreak in Arizona. [Journal_ref: SIAM J. Appl. Math. 82, 2036-2056 (2022)]
• Kinney, A. C., Current, S., & Lega, J. (2021). Aedes-AI: Neural network models of mosquito abundance. PLoS Computational Biology, 17(11), e1009467.
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  We present artificial neural networks as a feasible replacement for a mechanistic model of mosquito abundance. We develop a feed-forward neural network, a long short-term memory recurrent neural network, and a gated recurrent unit network. We evaluate the networks in their ability to replicate the spatiotemporal features of mosquito populations predicted by the mechanistic model, and discuss how augmenting the training data with time series that emphasize specific dynamical behaviors affects model performance. We conclude with an outlook on how such equation-free models may facilitate vector control or the estimation of disease risk at arbitrary spatial scales.
• Lega, J. C. (2021). Parameter Estimation from ICC curves. Journal of Biological Dynamics, 15, 195-212. doi:10.1080/17513758.2021.1912419
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  Incidence - Cumulative Cases (ICC) curves are introduced and shown to providea simple framework for parameter identification in the case of the mostelementary epidemiological model, consisting of susceptible, infected, andremoved compartments. This novel methodology is used to estimate the basicreproductive number of recent outbreaks, including the ongoing COVID-19epidemic.
• Lega, J., Brown, H. E., & Barrera, R. (2020). A 70% Reduction in Mosquito Populations Does Not Require Removal of 70% of Mosquitoes. Journal of Medical Entomology, 57(5), 1668-1670. doi:https://doi.org/10.1093/jme/tjaa066
• McGowan, C., Biggerstaff, M., Johansson, M., Apfeldorf, K. M., Ben-Nun, M., Brooks, L., Convertino, M., Erraguntla, M., Farrow, D. C., Freeze, J., Ghosh, S., Hyun, S., Kandula, S., Lega, J. C., Liu, Y., Michaud, N., Morita, H., Niemi, J., Ramakrishnan, N., , Ray, E. L., et al. (2019). Collaborative efforts to forecast seasonal influenza in the United States, 2015–2016. Scientific Reports, 9, 683. doi:https://doi.org/10.1038/s41598-018-36361-9
• Thompson, C., Saxberg, K., Lega, J. C., Tong, D., & Brown, H. E. (2019). A new gravity model for spatial interaction. Journal of Transport Geography, 79.
• Del Valle, S. Y., McMahon, B. H., Asher, J., Hatchett, R., Lega, J. C., Brown, H. E., Leany, M. E., Pantazis, Y., Roberts, D., Moore, S., Peterson, T., Escobar, L. E., Qiao, H., Hengartner, N. W., & Mukundan, H. (2018). Summary Results of the 2014-2015 DARPA Chikungunya Challenge. BMC Infectious Diseases, 18, 245. doi:http://dx.doi.org/10.1186/s12879-018-3124-7
• Ercolani, N. M., Kamburov, N., & Lega, J. C. (2018). The Phase Structure of Grain Boundaries. Philosophical Transactions of the Royal Society A, 376, 20170193. doi:http://dx.doi.org/10.1098/rsta.2017.0193
• Lega, J. C., Sethuraman, S., & Young, A. L. (2018). On collisions times of `self-sorting' interacting particles in one-dimension with random initial positions and velocities. Journal of Statistical Physics, 170, 1088-1122. doi:http://dx.doi.org/10.1007/s10955-018-1974-4
• Brown, H. E., Barrera, R., Comrie, A. C., & Lega, J. C. (2017). Effect of temperature thresholds on modeled Aedes aegypti population dynamics. Journal of Medical Entomology, 54(4), 869–877. doi:https://doi.org/10.1093/jme/tjx041
• Lega, J. C., Brown, H. E., & Barrera, R. (2017). Aedes aegypti (Diptera: Culicidae) abundance model improved with relative humidity and precipitation-driven egg hatching. Journal of Medical Entomology, 54(5), 1375–1384. doi:https://doi.org/10.1093/jme/tjx077
• Brubaker, N., & Lega, J. C. (2016). Capillary induced deformations of a thin elastic sheet. Philosophical Transactions of the Royal Society A, 374, 20150169. doi:10.1098/rsta.2015.0169
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  We develop a three-dimensional model for capillary origami systems in which a rectangular plate has finite thickness, is allowed to stretch and undergoes small deflections. This latter constraint limits our description of the encapsulation process to its initial folding phase. We first simplify the resulting system of equations to two dimensions by assuming that the plate has infinite aspect ratio, which allows us to compare our approach to known two-dimensional capillary origami models for inextensible plates. Moreover, as this two-dimensional model is exactly solvable, we give an expression for its solution in terms of its parameters. We then turn to the full three-dimensional model in the limit of small drop volume and provide numerical simulations showing how the plate and the drop deform due to the effect of capillary forces.
• Brubaker, N., & Lega, J. C. (2016). Two-dimensional capillary origami. Physics Letters A, 380, 83-87.
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  We describe a global approach to the problem of capillary origami that captures all equilibrium configurations in two-dimensional settings, with or without pinning of the liquid drop at the end points of the flexible membrane. We provide bifurcation diagrams showing the level of encapsulation of each equilibrium configuration as a function of the volume of liquid that it contains, as well as plots representing the energy of each equilibrium branch. Three different parameter regimes are identified, one of which predicts instantaneous encapsulation for small initial volumes of liquid.This article will be featured in Elsevier's 2017 Virtual Special Issue on Women in Physics.
• Lega, J. C., & Brown, H. E. (2016). Data-driven outbreak forecasting with a simple nonlinear growth model. Epidemics, 17, 19-26. doi:http://dx.doi.org/10.1016/j.epidem.2016.10.002
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  Recent events have thrown the spotlight on infectious disease outbreak response. We developed a data-driven method, EpiGro, which can be applied to cumulative case reports to estimate the order of magnitude of the duration, peak and ultimate size of an ongoing outbreak. It is based on a surprisingly simple mathematical property of many epidemiological data sets, does not require knowledge or estimation of disease transmission parameters, is robust to noise and to small data sets, and runs quickly due to its mathematical simplicity. Using data from historic and ongoing epidemics, we present the model. We also provide modeling considerations that justify this approach and discuss its limitations. In the absence of other information or in conjunction with other models, EpiGro may be useful to public health responders.
• Brown, H. E., Young, A., Lega, J. C., Andreadis, T. G., Schurich, J., & Comrie, A. C. (2015). Projection of Climate Change Influences on U.S. West Nile Virus Vectors. Earth Interactions, 19, 1-18. doi:http://dx.doi.org/10.1175/EI-D-15-0008.1
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  While quantitative estimates of the impact of climate change on health is an increasing concern for health care planners and climate change policy, the models to produce those estimates remain scarce. Herein, we describe a freely available dynamic simulation model parameterized for three West Nile virus vectors, which provides an effective tool for studying vector-borne disease risk due to climate change. The Dynamic Mosquito Simulation Model is parameterized with species specific temperature-dependent development and mortality rates. Using downscaled daily weather data, we estimate vector population dynamics under current and projected future climate scenarios for multiple locations across the country. Trends in mosquito abundance were variable by location, however, an extension of the vector activity periods, and by extension disease risk, was almost uniformly observed. Importantly, areas showing mid-summer decreases in vector abundance maybe off-set by shorter extrinsic incubation periods within the mosquito vector. Quantitative models of the effect of temperature on the virus and vector are critical to developing models of future disease risk.
• Brubaker, N. D., & Lega, J. C. (2015). Two-dimensional capillary origami with pinned contact line. SIAM Journal on Applied Mathematics, 75, 1275-1300.
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  To continue the move towards miniaturization in technology, developing new methods for fabricating micro- and nanoscale objects has become increasingly important. One potential method, called capillary origami, consists of placing a small drop of liquid on a thin, inextensible sheet. In this article, we model the static configurations of this system in an idealized two-dimensional setting and describe how the plate will fold due to capillary forces. We do this by minimizing the total energy of the system, which consists of bending and interfacial components whose relative importance can be measured by a dimensionless parameter $\lambda$. The deflection of the plate is characterized in terms of bifurcation diagrams, where the bifurcation parameter is the drop's size. This allows us to consider the quasi-static evolution of the system in the presence of evaporation. Variations in this bifurcation diagram for various $\lambda$ are then studied, leading us to organize the physical description of the system's behavior into three different regimes. The present approach provides a general framework for the study of capillary origami that can be extended to three-dimensional settings.
• Lindsay, A. E., Lega, J. C., & Glasner, K. B. (2015). Regularized Model of Post-Touchdown Configurations in Electrostatic MEMS: Interface Dynamics. The IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv011, 29.
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  Interface dynamics of post contact states in regularized models of electrostatic-elastic interactions are analyzed. A canonical setting for our investigations is the field of Micro-Electromechanical Systems (MEMS) in which flexible elastic structures may come into physical contact due to applied Coulomb forces. We study the dynamic features of a recently derived regularized model (A.E. Lindsay et al, Regularized Model of Post-Touchdown Configurations in Electrostatic MEMS: Equilibrium Analysis, Physica D, 2014), which describes the system past the quenching singularity associated with touchdown, that is after the components of the device have come together. We build on our previous investigations of steady-state solutions by describing how the system relaxes towards these equilibria. This is accomplished by deriving a reduced dynamical system that governs the evolution of the contact set, thereby providing a detailed description of the intermediary dynamics associated with this bistable system. The analysis yields important practical information on the timescales of equilibration.
• Lega, J. C., Buxner, S., Blonder, B., & Tama, F. (2014). Explorations in integrated science. Journal of College Science Teaching, 43(4), 24-29.
• Lindsay, A. E., Lega, J., & Glasner, K. B. (2014). Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis. PHYSICA D: NONLINEAR PHENOMENA, 280, 95-108.
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  In canonical models of Micro-Electra Mechanical Systems (MEMS), an event called touchdown whereby the electrical components of the device come into contact, is characterized by a blow up in the governing equations and a non-physical divergence of the electric field. In the present work, we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria. We employ techniques from variational calculus, dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations. (C) 2014 Elsevier B.V. All rights reserved.
• Lega, J. (2013). Erratum: Collective behaviors in two-dimensional systems of interacting particles (SIAM Journal on Applied Dynamical Systems (2011) 10 (1213-1231)). SIAM Journal on Applied Dynamical Systems, 12(4), 2093-.
• Lindsay, A. E., Lega, J., & Sayas, F. J. (2013). The quenching set of a MEMS capacitor in two-dimensional geometries. Journal of Nonlinear Science, 23(5), 807-834.
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  Abstract: The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro-Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity. © 2013 Springer Science+Business Media New York.
• Moulton, D. E., & Lega, J. (2013). Effect of disjoining pressure in a thin film equation with non-uniform forcing. European Journal of Applied Mathematics, 24(6), 887-920.
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  Abstract: We explore the effect of disjoining pressure on a thin film equation in the presence of a non-uniform body force, motivated by a model describing the reverse draining of a magnetic film. To this end, we use a combination of numerical investigations and analytical considerations. The disjoining pressure has a regularizing influence on the evolution of the system and appears to select a single steady-state solution for fixed height boundary conditions; this is in contrast with the existence of a continuum of locally attracting solutions that exist in the absence of disjoining pressure for the same boundary conditions. We numerically implement matched asymptotic expansions to construct equilibrium solutions and also investigate how they behave as the disjoining pressure is sent to zero. Finally, we consider the effect of the competition between forcing and disjoining pressure on the coarsening dynamics of the thin film for fixed contact angle boundary conditions. Copyright © Cambridge University Press 2013.
• Lindsay, A. E., & Lega, J. (2012). Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a mems capacitor. SIAM Journal on Applied Mathematics, 72(3), 935-958.
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  Abstract: Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric two-dimensional case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity. © 2012 Society for Industrial and Applied Mathematics.
• Herrera-Valdez, M. A., & Lega, J. (2011). Reduced models for the pacemaker dynamics of cardiac cells. Journal of Theoretical Biology, 270(1), 164-176.
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  PMID: 20932980;Abstract: We introduce three- and two-dimensional biophysical models of cardiac excitability derived from a 14-dimensional model of the sinus venosus [Rasmusson, R., et al., 1990. Am. J. Physiol. 259, H352-369]. The reduced models capture normal pacemaking dynamics with a small complement of ionic currents. The two-dimensional model bears some similarities with the Morris-Lecar model [Morris, C., Lecar, H., 1981. Biophysical Journal, 35, 193-213]. Because they were reduced from a biophysical model, both models depend on parameters that were obtained from experimental data. Even though the correspondence with the original model is not exact, parameters may be adjusted to tune the reductions to fit experimental traces. As a consequence, unlike other generic low-dimensional models, the models introduced here provide a means to relate physiologically relevant characteristics of pacemaker potentials such as diastolic depolarization, plateau, and action potential frequency, to biophysical variables such as the relative abundance of membrane channels and channel kinetic rates. In particular, these models can lead to an explicit description of how the shape of cardiac action potentials depends on the relative contributions and states of inward and outward currents. By being physiologically derived and computationally efficient, the models presented in this article are useful tools for theoretical studies of excitability at the cellular and network levels. © 2010.
• Lafortune, S., Lega, J., & Madrid, S. (2011). Instability of local deformations of an elastic rod: Numerical evaluation of the evans function. SIAM Journal on Applied Mathematics, 71(5), 1653-1672.
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  Abstract: We present a method for the numerical evaluation of the Evans function that doesnot require integration in an associated exterior algebra space. This technique is suitable for thedetection of bifurcations and is particularly useful when the dimension of the linearized systemand/or the dimension of the converging subspaces at infinity is large. We test this approach byinvestigating the stability of a two-parameter family of traveling pulse solutions to two coupledKlein-Gordon equations. The spectral stability of these pulses is completely understood analytically [S. Lafortune and J. Lega, SIAM J. Math. Anal. , 36 (2005), pp. 1726-1741], and we show that ournumerical method is able to detect bifurcations of the pulse family with very good accuracy. © 2011 Society for Industrial and Applied Mathematics.
• Lega, J. (2011). Collective behaviors in two-dimensional systems of interacting particles. SIAM Journal on Applied Dynamical Systems, 10(4), 1213-1231.
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  Abstract: This article presents results of molecular dynamics simulations that show the emergence of collective behaviors in a two-dimensional system of particles (hard disks) interacting through a properly chosen collision rule. The particles, which are of finite size and are in free flight between collisions, are not self-propelled. They tumble randomly like bacteria and interact only when they collide, not through continuous potential forces. This work therefore indicates that interactions at the microscopic level, which occur only locally and discretely both in time and space, are sufficient to lead to large-scale macroscopic behaviors. Order parameters that capture and quantify the formation of collective behaviors are introduced and used to describe how the choice of collision rule affects the steady state dynamics of the system, by comparing the outcome to the standard case of elastic collisions. This work was motivated by recent results on the dynamics of bacterial colonies. Possible applications of the present approach to other systems are also discussed. © 2011 Society for Industrial and Applied Mathematics.
• Moulton, D. E., & Lega, J. (2009). Reverse draining of a magnetic soap film - Analysis and simulation of a thin film equation with non-uniform forcing. Physica D: Nonlinear Phenomena, 238(22), 2153-2165.
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  Abstract: We analyze and classify equilibrium solutions of the one-dimensional thin film equation with no-flux boundary conditions and in the presence of a spatially dependent external forcing. We prove theorems that shed light on the nature of these equilibrium solutions, guarantee their validity, and describe how they depend on the properties of the external forcing. We then apply these results to the reverse draining of a one-dimensional magnetic soap film subject to an external non-uniform magnetic field. Numerical simulations illustrate the convergence of the solutions towards equilibrium configurations. We then present bifurcation diagrams for steady state solutions. We find that multiple stable equilibrium solutions exist for fixed parameters, and uncover a rich bifurcation structure to these solutions, demonstrating the complexity hidden in a relatively simple looking evolution equation. Finally, we provide a simulation describing how numerical solutions traverse the bifurcation diagram, as the amplitude of the forcing is slowly increased and then decreased. © 2009 Elsevier B.V. All rights reserved.
• Lega, J., & Passot, T. (2007). Hydrodynamics of bacterial colonies. Nonlinearity, 20(1), C1-C16.
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  Abstract: Understanding the growth and dynamics of bacterial colonies is a fascinating problem, which requires combining ideas from biology, physics and applied mathematics. We briefly review the recent experimental and theoretical literature relevant to this question and describe a hydrodynamic model (Lega and Passot 2003 Phys. Rev. E 67 031906, 2004 Chaos 14 562-70), which captures macroscopic motions within bacterial colonies, as well as the macroscopic dynamics of colony boundaries. The model generalizes classical reaction-diffusion systems and is able to qualitatively reproduce a variety of colony shapes observed in experiments. We conclude by listing open questions about the stability of interfaces as modelled by reaction-diffusion equations with nonlinear diffusion and the coupling between reaction-diffusion equations and a hydrodynamic field. © 2007 IOP Publishing Ltd and London Mathematical Society.
• Lafortune, S., & Lega, J. (2005). Spectral stability of local deformations of an elastic rod: Hamiltonian formalism. SIAM Journal on Mathematical Analysis, 36(6), 1726-1741.
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  Abstract: Hamiltonian methods are used to obtain a necessary and sufficient condition for the spectral stability of pulse solutions to two coupled nonlinear Klein-Gordon equations. These equations describe the near-threshold dynamics of an elastic rod with circular cross section. The present work completes and extends a recent analysis of the authors' [Phys. D, 182 (2003), pp. 103-124], in which a sufficient condition for the instability of "nonrotating" pulses was found by means of Evans function techniques. © 2005 Society for Industrial and Applied Mathematics.
• Lega, J., & Passot, T. (2004). Hydrodynamics of bacterial colonies: Phase diagrams. Chaos, 14(3), 562-570.
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  PMID: 15446966;Abstract: We present numerical simulations of a recent hydrodynamic model describing the growth of bacterial colonies on agar plates. We show that this model is able to qualitatively reproduce experimentally observed phase diagrams, which relate a colony shape to the initial quantity of nutrients on the plate and the initial wetness of the agar. We also discuss the principal features resulting from the interplay between hydrodynamic motions and colony growth, as described by our model. © 2004 American Institute of Physics.
• Lega, J., & Passot, T. (2004). Inverse cascade and energy transfer in forced low-Reynolds number two-dimensional turbulence. Fluid Dynamics Research, 34(5), 289-297.
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  Abstract: Using numerical simulations of the forced two-dimensional Navier-Stokes equation, it is shown that the amount of energy transferred to large scales is related to the Reynolds number in a unique fashion. It is also observed that the critical value of the initial Reynolds number for the onset of an inverse cascade is lowered as the scale of the forcing approaches the size of the system, or in the presence of anisotropy. This study is motivated by recent experiments with bacterial colonies, and their description in terms of a hydrodynamic model. © 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.
• Lafortune, S., & Lega, J. (2003). Instability of local deformations of an elastic rod. Physica D: Nonlinear Phenomena, 182(1-2), 103-124.
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  Abstract: We study the instability of pulse solutions of two coupled non-linear Klein-Gordon equations by means of Evans function techniques. The system of coupled Klein-Gordon equations considered here describes the near-threshold dynamics of a three-dimensional elastic rod with circular cross-section, subject to constant twist. We determine a condition on the speed of the traveling pulse which ensures spectral instability. © 2003 Elsevier Science B.V. All rights reserved.
• Lega, J., & Passot, T. (2003). Hydrodynamics of bacterial colonies: A model. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 67(3 1), 031906/1-031906/8.
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  PMID: 12689100;Abstract: A hydrodynamic model that gives a general description of bacterial colonies growing in soft agar plates is proposed. This scheme provides a framework in which macroscopic reaction-diffusion models of bacterial colonies are justified on the basis of hydrodynamic considerations. With the given model, colonies that are drier in the interior than at the boundary can be described.
• Lega, J. (2001). Traveling hole solutions of the complex Ginzburg-Landau equation: A review. Physica D: Nonlinear Phenomena, 152-153, 269-287.
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  Abstract: This paper reviews recent works on localized solutions of the one-dimensional complex Ginzburg-Landau (CGL) equation known as traveling holes. Such coherent structures seem to play an important role in the disordered dynamics displayed by CGL at a finite distance past the Benjamin-Feir instability threshold. We discuss these objects in the broader context of weak turbulence and summarize some of their properties. © 2001 Elsevier Science B.V.
• Lega, J., & Goriely, A. (1999). Pulses, fronts and oscillations of an elastic rod. Physica D: Nonlinear Phenomena, 132(3), 373-391.
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  Abstract: Two coupled nonlinear Klein-Gordon equations modeling the three-dimensional dynamics of a twisted elastic rod near its first bifurcation threshold are analyzed. First, it is shown that these equations are Hamiltonian and that they admit a two-parameter family of traveling wave solutions. Second, special solutions corresponding to simple deformations of the elastic rod are considered. The stability of such configurations is analyzed by means of two coupled nonlinear Schrödinger equations, which are derived from the nonlinear Klein-Gordon equations in the limit of small deformations. In particular, it is shown that periodic solutions are modulationally unstable, which is consistent with the looping process observed in the writhing instability of elastic filaments. Third, numerical simulations of the nonlinear Klein-Gordon equations suggesting that traveling pulses are stable, are presented. © 1999 Elsevier Science B.V. All rights reserved.
• Lega, J., & Mendelson, N. H. (1999). Control-parameter-dependent Swift-Hohenberg equation as a model for bioconvection patterns. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 59(6), 6267-6274.
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  PMID: 11969610;Abstract: We consider a complex Swift-Hohenberg equation with control-parameter-dependent coefficients and use it as a mode to describe dynamical features seen in an experimental bacterial bioconvecti on pattern. In particular, we give numerical results showing the development of a phase-unstable pattern behind a moving front. ©1999 The American Physical Society.
• Bottin, S., & Lega, J. (1998). Pulses of tunable size near a subcritical bifurcation. European Physical Journal B, 5(2), 299-308.
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  Abstract: We show that a nonlinear gradient term can be used to tune the width of pulse-like solutions to a generalized quintic Ginzburg-Landau equation. We investigate the dynamics of these solutions and show that weakly turbulent patches can persist for long times. Analogies with turbulent spots in plane Couette flows are discussed.
• Mendelson, N. H., & Lega, J. (1998). A complex pattern of traveling stripes is produced by swimming cells of Bacillus subtilis. Journal of Bacteriology, 180(13), 3285-3294.
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  PMID: 9642178;PMCID: PMC107280;Abstract: Motile cells of Bacillus subtilis inadvertently escaped from the surface of an agar disk that was surrounded by a fluid growth medium and formed a migrating population in the fluid. When viewed from above, the population appeared as a cloud advancing unidirectionally into the fresh medium. The cell population became spontaneously organized into a series of stripes in a region behind the advancing cloud front. The number of stripes increased progressively until a saturation value of stripe density per unit area was reached. New stripes arose at a fixed distance behind the cloud front and also between stripes. The spacing between stripes underwent changes with time as stripes migrated towards and away from the cloud front. The global pattern appeared to be stretched by the advancing cloud front. At a time corresponding to approximately two cell doublings after pattern formation, the pattern decayed, suggesting that there is a maximum number of cells that can be maintained within the pattern. Stripes appear to consist of high concentrations of cells organized in sinking columns that are part of a bioconvection system. Their behavior reveals an interplay between bacterial swimming, bioconvection-driven fluid motion, and cell concentration. A mathematical model that reproduces the development and dynamics of the stripe pattern has been developed.
• Calderón, O. G., Pérez-Garcí, V. M., Lega, J., & Guerra, J. M. (1997). Loss-induced transverse effects in lasers. Optics Communications, 143(4-6), 315-321.
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  Abstract: We analyze the effect of losses on transverse patterns of a single longitudinal mode laser. In particular, we find a diffusion term which predicts the existence of a diffusive cutoff in the transverse spectrum. The linear stability analysis of the non-lasing solution shows that the detuning value (Δ = 0), that separates the two types of solutions above threshold, shifts to a positive value, which is important for lasers with a high polarization dephasing rate (γ⊥) such as Class B lasers. © 1997 Elsevier Science B.V.
• Hochheiser, D., Moloney, J. V., & Lega, J. (1997). Controlling optical turbulence. Physical Review A - Atomic, Molecular, and Optical Physics, 55(6), R4011-R4014.
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  Abstract: A robust global control strategy, implemented as a spatial filter with delayed feedback, is shown to stabilize and steer the weakly turbulent output of a spatially extended system. The latter is described by a generalized complex Swift-Hohenberg equation [J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994); Physica D 83, 478 (1995)], which is used as a generic model for pattern formation in the transverse section of semiconductor lasers. Our technique is particularly adapted to optical systems and should provide convenient experimental control of filamentation in wide-aperture lasers. [S1050-2947(97)50206-X].
• Lega, J., & Fauve, S. (1997). Traveling hole solutions to the complex Ginzburg-Landau equation as perturbations of nonlinear Schrödinger dark solitons. Physica D: Nonlinear Phenomena, 102(3-4), 234-252.
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  Abstract: We describe complex Ginzburg-Landau (CGL) traveling hole solutions as singular perturbations of nonlinear Schrödinger (NLS) dark solitons. Modulation of the free parameters of the NLS solutions leads to a dynamical system describing the CGL dynamics in the vicinity of a traveling hole solution. © 1997 Elsevier Science B.V. All rights reserved.
• Harkness, G. K., Lega, J., & Oppo, G. (1996). Measuring disorder with correlation functions of averaged patterns. Physica D: Nonlinear Phenomena, 96(1-4), 26-29.
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  Abstract: We propose an indicator of disorder which can be measured on averaged intensity images of a weakly turbulent complex field. Copyright © 1996 Elsevier Science B.V. All rights reserved.
• Harkness, G. K., Lega, J., & Oppo, G. -. (1996). Travelling wave patterns in lasers with curved mirrors. Technical Digest - European Quantum Electronics Conference, 42-.
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  Abstract: The study of pattern formation in the transverse section of lasers with plane cavity mirrors shows that the fundamental solutions are transverse traveling waves. On the other hand, lasers with curved cavity mirrors generate outputs which look like a combination of empty cavity modes. The mechanism behind the selection of travelling waves and cavity modes was found to be analogous. Numerical simulations show that for large input energy (pump), patterns of travelling wave nature are recovered even the mirrors are curved. As the radius of curved mirrors decreases the threshold value of the pump increase.
• Lega, J., & Vince, J. (1996). Temporal forcing of traveling wave patterns. Journal de Physique I, 6(11), 1417-1434.
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  Abstract: We experimentally and numerically study one-dimensional, temporally forced wave patterns and analyze the transition from unforced traveling waves to forced standing waves when a source defect is present in the system. Our control parameter is the amplitude of the forcing. Two scenarios are identified, depending on the distance from the bifurcation threshold of traveling waves. © Les Éditions de Physique 1996.
• Harkness, G. K., Lega, J. C., & Oppo, G. (1994). Correlation functions in the presence of optical vortices. Chaos, Solitons and Fractals, 4(8-9), 1519-1533.
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  Abstract: We evaluate two-dimensional field and intensity correlation functions for the optical Ginzburg-Landau and laser equations in the presence of optical vortices. Configurations of rotating vortices, vortices superimposed on travelling waves and defect mediated turbulance are analysed and compared. Intensity correlations provide a qualitative indicator for the degree of disorder of spatio-temporal evolutions. For example, patterns with few rotating vortices have correlation lengths comparable to or larger than the transverse size of the system, while few vortices superimposed on travelling waves can generate weak turbulance. © 1994.
• Daviaud, F., Lega, J., Bergé, P., Coullet, P., & Dubois, M. (1992). Spatio-temporal intermittency in a 1D convective pattern: Theoretical model and experiments. Physica D: Nonlinear Phenomena, 55(3-4), 287-308.
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  Abstract: We describe the occurence of spatio-temporal intermittency in a one-dimensional convective system that first shows time-dependent patterns. We recall experimental results and propose a model based on the normal form description of a secondary Hopf bifurcation of a stationary periodic structure. Numerical simulations of this model show spatio-temporal intermittent behaviors, which we characterize briefly and compare to those given by the experiment. © 1992.
• Lega, J., Janiaud, B., Jucquois, S., & Croquette, V. (1992). Localized phase jumps in wave trains. Physical Review A, 45(8), 5596-5604.
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  Abstract: We show experimental evidence of traveling hole defects, which are reminiscent of analytical solutions to complex Ginzburg-Landau equations. Also, these objects seem to play an important role in the development of phase instability of oscillatory patterns. © 1992 The American Physical Society.
• Coullet, P., Lega, J., & Pomeau, Y. (1991). Dynamics of Bloch Walls in a Rotating Magnetic Field: A Model. EPL, 15(2), 221-226. doi:10.1209/0295-5075/15/2/019
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  We both analytically and numerically show the existence of a drift of Bloch walls when submitted to a uniform parallel-to-the-wall-plane rotating magnetic field. The drift velocity changes sign with Bloch wall handedness and is proportional to the amplitude square of the magnetic field, when the latter is small.
• Lega, J. (1991). Defect-mediated turbulence. Computer Methods in Applied Mechanics and Engineering, 89(1-3), 419-424.
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  Abstract: We summarize recent results about a dynamic regime observed in numerical simulations of two-dimensional Ginzburg-Landau equations, for which defects are spontaneously created in the system and are responsible for its disorganization. © 1991.
• Ciliberto, S., Coullet, P., Lega, J., Pampaloni, E., & Perez-Garcia, C. (1990). Defects in Roll-Hexagon competition. Physical Review Letters, 65(19), 2370-2373.
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  Abstract: The defects of a system where hexagons and rolls are both stable solutions are considered. On the basis of topological arguments we show that the unstable phase is present in the core of the defects. This means that a roll is present in the penta-hepta defect of hexagons and that a hexagon is found in the core, of a grain boundary connecting rolls with different orientations. These results are verified in an experiment of thermal convection under non-Boussinesq conditions. © 1990 The American Physical Society.
• Coullet, P., Lega, J., Houchmanzadeh, B., & Lajzerowicz, J. (1990). Breaking chirality in nonequilibrium systems. Physical Review Letters, 65(11), 1352-1355.
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  Abstract: At equilibrium, Bloch walls are chiral interfaces between domains with different magnetisation. Far from equilibrium, a set of forced oscillators can exhibit walls between states with different phases. In this Letter, we show that when these walls become chiral. they move with a velocity simply related to their chirality. This surprising behavior is a straightforward consequence of nonvariational effects. which are typical of nonequilibrium systems.
• Gil, L., Lega, J., & Meunier, J. L. (1990). Statistical properties of defect-mediated turbulence. Physical Review A, 41(2), 1138-1141.
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  Abstract: We study some statistical properties of a turbulent state described by a generalized Ginzburg-Landau equation and characterized by topological defects. © 1990 The American Physical Society.
• Coullet, P., Gil, L., & Lega, J. (1989). A form of turbulence associated with defects. Physica D: Nonlinear Phenomena, 37(1-3), 91-103.
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  Abstract: We show by means of numerical simulations of complex Ginzburg-Landau equations that phase instability leads to the spontaneous nucleation of topological defects, which disorganize the system. © 1989.
• Coullet, P., Gil, L., & Lega, J. (1989). Defect-mediated turbulence. Physical Review Letters, 62(14), 1619-1622.
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  Abstract: We describe a turbulent state characterized by the presence of topological defects. This topological turbulence is likely to be experimentally observed in nonequilibrium systems. © 1989 The American Physical Society.
• Coullet, P., Gil, L., & Lega, J. (1989). Une forme de turbulence associée aux défauts topologiques. Mathematical Modelling and Numerical Analysis, 23(3), 385-394. doi:10.1051/m2an/1989230303851
• Coullet, P., & Lega, J. (1988). Defect-mediated turbulence in wave patterns. EPL, 7(6), 511-516. doi:10.1209/0295-5075/7/6/006
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  A turbulent behaviour of wave patterns is described, which is related to the presence of dislocations. By means of numerical simulations of 2D complex Ginzburg-Landau equations, it is shown that phase instability leads in spatially extended systems to spontaneous nucleation of topological defects. The appearance of these localized amplitude perturbations is interpreted as the consequence of the revolt of the slaved amplitude modes. Once created, those defects move through the system and break the order induced by the wave pattern. The resulting turbulent state has been termed "topological turbulence".
• Coullet, P., Elphick, C., Gil, L., & Lega, J. (1987). Topological defects of wave patterns. Physical Review Letters, 59(8), 884-887.
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  Abstract: We identify the defects of waves by means of topological arguments and study them in the framework of Landau-type analysis. It is shown that they correspond to sinks, sources, or dislocations of traveling waves, and to dislocations of standing waves. © 1987 The American Physical Society.

Presentations

• Lega, J. C. (2023, August). A dynamical systems approach to map enumeration. 10th International Congress on Industrial and Applied Mathematics, Session on Painlevé equations, Applications, and Related Topics. Tokyo, Japan - Delivered online.
• Lega, J. C. (2022, April). Dynamical systems properties of the Freud recurrence. Session on Discrete Painleve Equations and Related Topics, 12th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory. Athens Georgia.
• Lega, J. C. (2022, July). Models of Mosquito Abundance. ASU Summer REU Colloquium Series. Arizona State University.
• Lega, J. C. (2022, October). On the Number of Quadrangulations of a Topological Surface. Mathematics Colloquium, University of Houston.
• Lega, J. C. (2021, February 17). Modeling and Forecasting the Spread of the Pandemic. Webinar Series on COVID-19: Breaking and Raising Boundaries. Online: International research laboratory CNRS/ENS-PSL, COVIDAM Blog, and Institut des Amériques.
• Lega, J. C. (2021, February 24). Modeling in the time of the pandemic. Mathematics Colloquium, Southern Methodist University. Online: Department of Mathematics, Southern Methodist University, Dallas, TX.
• Lega, J. C. (2021, June). A New Take on Outbreak Dynamics. Minisymposium on Evolutionary Theory of Disease, Virtual SMB 2021 Annual Meeting. Online: Society for Mathematical Biology.
• Lega, J. C. (2021, May). Forecasting Disease Risk and Spread. French American Innovation Days: Facing the Predictably Unpredictable. Online: Ambassade de France aux Etats Unis.
• Lega, J. C. (2021, October 26). A dynamical systems view of special solutions to the discrete Painlevé I equation. Nonlinear Waves and Coherent Structures Webinar. Online: UMass Amherst, Bowdoin College, Cal Poly, San Luis Obispo.
• Lega, J. C. (2020, October). Epidemiological Forecasting with ICC curves and data assimilation. ASU Mathbio SeminarArizona State University.
• Lega, J. C. (2020, October). Epidemiological Forecasting with Simple Nonlinear Models. Dynamical Systems Seminar, University of MinnesotaUniversity of Minnesota.
• Lega, J. C. (2020, October). Phase singularities and defects in the Swift-Hohenberg equation. Fall Western AMS Sectional meeting, Special Session on “Free boundary problems arising in applications”. University of Utah, Salt Lake City, Utah: AMS.
• Lega, J. C. (2019, April). Grain boundaries of the Swift-Hohenberg equation: simulations and analysis. 11th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory. Athens, GA.
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  Special Session on Nonlinear Evolutionary Equations: Theory, Numerics and Experiments.
• Lega, J. C. (2019, August). Panel: Moving to More Useful Forecasts at the State/Local Level: The Forecasters Perspective. FluSight Seasonal Influenza Forecasting Workshop. Atlanta, GA: Council of State and Territorial Epidemiologists.
• Lega, J. C. (2019, July). Phase singularities and defects in pattern forming systems. 9th International Congress on Applied Mathematics. Valencia, Spain.
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  Mini-symposium on Existence and stability of nonlinear waves
• Lega, J. C. (2019, March). Panel: Mathematical Perspectives and Vision for Preparation and Pathways in Mathematical Modeling. Critical Issues in Mathematics Education 2019: Mathematical Modeling in K-16: Community and Cultural Context. Berkeley, CA: MSRI Critical Issues in Mathematics Education.
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  Presentation (with slides) as part of a panel on Pathways in Mathematical Modeling.
• Lega, J. C. (2019, March). Transdisciplinary modeling of mosquito-borne diseases. Southwestern Undergraduate Mathematics Research Conference (SUnMaRC). Tucson, Arizona: University of Arizona.
• Lega, J. C. (2018, August). Forecasting the Flu with Simple Nonlinear Models. CSTE/CDC Seasonal Influenza Forecasting Workshop. Atlanta, GA: CDC & Council of State and Territorial Epidemiologists.
• Lega, J. C. (2018, March). Trans-disciplinary modeling of mosquito-borne diseases. Modeling and Computation Seminar, University of Arizona.
• Lega, J. C. (2018, November). Modeling the spread of vector-borne diseases on regional transportation networks. 2018 ESA, ESC, and ESBC Joint Annual Meeting. Vancouver, BC, Canada, November 11 – 14, 2018.
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  Invited presentation at the MUVE Section Symposium on Predicting Vector-Borne Diseases Spread in Changing Natural and Social Landscapes.
• Lega, J. C. (2017, April). Forecasting the Flu. Uncertainty Quantification Seminar, University of Arizona. UA Department of Mathematics.
• Lega, J. C. (2017, April). The phase structure of grain boundaries. IIMAS Applied Mathematics Colloquium, UNAM. Mexico City: UNAM (Universidad Nacional Autónoma de México) / IIMAS (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas).
• Lega, J. C. (2017, April). Three models to help understand the spread of mosquito-borne diseases. 10th IIMAS Colloquium, UNAM. Mexico City: UNAM (Universidad Nacional Autónoma de México) / IIMAS (Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas).
• Lega, J. C. (2017, August). Forecasting the Flu with Simple Nonlinear Models. Seasonal Influenza Forecasting Workshop, CDC. Atlanta, GA: Centers for Disease Control and Prevention.
• Lega, J. C. (2017, February). Capillary Origami. Analysis, Dynamics, and Applications Seminar, University of Arizona.
• Lega, J. C. (2017, March). The phase structure of grain boundaries. 1126 AMS Meeting. Charleston, SC.
• Lega, J. C. (2017, May). Patterns, defects, and phase singularities. Rocky Mountain Partial Differential Equations Conference. Provo, UT: National Science Foundation, Brigham-Young University.
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  Plenary speaker
• Lega, J. C. (2017, May). The Phase Structure of Grain Boundaries. 2017 SIAM Conference on Applications of Dynamical Systems. Snowbird, UT: SIAM.
• Lega, J. C. (2017, November). A Three-pronged Approach to Predicting the Spread of Mosquito-borne Diseases. Mathematics Colloquium, Colorado State University. Fort Collins, CO: Colorado State University.
• Lega, J. C. (2017, October). Patterns, defects, and phase singularities. Analysis, Dynamics, and Applications Seminar, University of Arizona.
• Lega, J. C. (2016, 04-12-2016). Models for mosquito abundance and infectious disease outbreak forecasting. Quantitative Biology Colloquium, University of Arizona.
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  Title: Models for mosquito abundance and infectious disease outbreak forecastingAbstract: In this talk, I will present an agent-based model for mosquito abundance that uses temperature and precipitation as time-varying parameters. I will show results that describe the effect of climate change on mosquito abundance and discuss the importance of how rainfall is taken into account in the model. I will then move to a very simple macroscopic description for the spread of infectious diseases and illustrate applications of this model to various outbreaks, including the recent chikungunya epidemic in the Americas. This work is in collaboration with Heidi Brown (College of Public Health) and many other researchers on and off campus.
• Lega, J. C. (2016, August 8-11). Defects in the Swift-Hohenberg Equation. 2016 SIAM Conference on Nonlinear Waves and Coherent Structures. Philadelphia, PA.
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  Talk in Session on Existence and Stability of Nonlinear Waves and Patterns.Abstract: I will discuss static and dynamic properties of grain boundaries in pattern-forming systems, using the Swift-Hohenberg equation as a canonical model. In particular, I will focus on the transition between grain boundaries, pairs of concave-convex disclinations, and dislocations. I will present a mix of numerical simulations and analytical results. Some of this work is joint with N. Ercolani and N. Kamburov (University of Arizona).
• Lega, J. C. (2016, August-September). Flu forecasting with EpiGro. CDC Seasonal Influenza Forecasting Workshop. Centers for Disease Control and Prevention, Atlanta, GA: Centers for Disease Control and Prevention.
• Ercolani, N. M., Kamburov, N. A., & Lega, J. C. (2015, December). How Defects are Born. SIAM conference on Analysis of Partial Differential Equations. Scottsdale, AZ.
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  Pattern-forming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems invariant under translations, disclinations in stripe-forming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized, phase gradients.This talk will describe recent advances on a long-term project whose goal is to follow such a defect-forming mechanism in a system that is amenable to analysis. Specifically, we focus on the appearance of pairs of dislocations at the core of a grain boundary of the Swift-Hohenberg equation. Taking advantage of the variational nature of this system, we show that as the angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the Cross-Newell equation, develops large derivatives in a region of diminishing size.This is joint work with N. Ercolani and N. Kamburov.
• Lega, J. C. (2015, April). How Defects are Born. The 1st Annual Meeting of SIAM Central States Section. Rolla, M.
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  Pattern-forming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems invariant under translations, disclinations in stripe-forming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized, phase gradients.This talk will describe recent advances on a long-term project whose goal is to follow such a defect-forming mechanism in a system that is amenable to analysis. Specifically, we focus on the appearance of pairs of dislocations at the core of a grain boundary of the Swift-Hohenberg equation. Taking advantage of the variational nature of this system, we show that as the angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the Cross-Newell equation, develops large derivatives in a region of diminishing size.This is joint work with N. Ercolani and N. Kamburov.
• Lega, J. C. (2015, April). Mathematical Modeling & Applications to Public Health. EPI Seminar, UA College of Public Health.
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  EPI seminar, College of Public Health, University of Arizona, 29 April, 2015
• Lega, J. C. (2015, May). Physica D: Nonlinear Phenomena. SIAM Conference on Applications of Dynamical Systems - Journal Editors Panel and Reception. Snowbird, Utah: SIAM.
• Lega, J. C. (2015, November). Capillary Origami. Mathematics Colloquium, University of Central Florida. University of Central Florida: UCF.
• Lega, J. C. (2015, November). Explorations in Undergraduate Education. Seminar, University of Central Florida. University of Central Florida: University of Central Florida.
• Lega, J. C., & Brown, H. E. (2015, May). Modeling the Spread of Chikungunya in the Caribbean and Central America. DARPA Chikungunya Challenge Finale. DARPA: DARPA.
• Lega, J. C., & Brown, H. E. (2015, October). Modeling the Spread of Chikungunya in the Caribbean and Central America. UA Microlunch SeriesUniversity of Arizona.
• Lega, J. C., Ercolani, N. M., & Kamburov, N. A. (2015, April). How Defects are Born. The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory.
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  Pattern-forming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems invariant under translations, disclinations in stripe-forming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized, phase gradients.This talk will describe recent advances on a long-term project whose goal is to follow such a defect-forming mechanism in a system that is amenable to analysis. Specifically, we focus on the appearance of pairs of dislocations at the core of a grain boundary of the Swift-Hohenberg equation. Taking advantage of the variational nature of this system, we show that as the angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the Cross-Newell equation, develops large derivatives in a region of diminishing size.
• Lega, J. C., Ercolani, N. M., & Kamburov, N. A. (2014, July). Grain boundaries of the Swift-Hohenberg and regularized Cross-Newell equations. Special session on Traveling Waves and Patterns, 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications. Madrid, Spain.
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  Grain boundaries in extended two-dimensional pattern forming systems are curves separating regions of slanted rolls. When the angle between the rolls in each of the two regions exceeds a certain threshold, it is known [1,2] that the core of the grain boundary transforms into a chain of convex-concave disclinations. Even though the regularized Cross-Newell (RCN) phase diffusion equation cannot describe this transition all the way to the appearance of defects, it can nevertheless be used to address the question of whether the transition results from an instability of the grain boundary core, and if so, to describe this instability. To this end, we will take full advantage of the existence of an exact grain-boundary solution of RCN and of the variational nature of this equation. I will also show numerical simulations and connect our results to those of Haragus and Scheel [3] on grain boundaries of the Swift-Hohenberg equation.References[1] N.M. Ercolani, R. Indik, A.C. Newell, and T. Passot, J. Nonlinear Sci. 10, 223-274 (2000).[2] N.M. Ercolani and S.C. Venkataramani, J. Nonlinear Sci. 19, 267-300 (2009).[3] M. Haragus and A. Scheel, European Journal of Applied Mathematics 23, 737-759 (2012).
• Lega, J. C. (2011, November 4-6). Collective Behaviors in Two-dimensional Systems of Interacting Particles and Rods. Geometric Methods for Innite-Dimensional Dynamical Systems. Brown University, Providence, RI: Brown University, NSF.
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  A meeting in celebration of Chris Jones' 60th birthday, Brown University

Case Studies

• Roach, M., Brown, H. E., Clark, R., Hondula, D., Lega, J. C., Rabby, Q., Schweers, N., & Tabor, J. (2017. Projections of Climate Impacts on Vector-Borne Diseases and Valley Fever in Arizona(p. 20).